The purpose of this study is to test any good and bad effects of the study drug called Abemaciclib. Abemaciclib could shrink your cancer but it could also cause side effects. Researchers hope to learn if the study drug will delay the growth of the cancer or shrink the cancer by at least one quarter compared to its present size. Abemaciclib is not FDA approved and has not been tested in liposarcoma, but it has shrunk tumors in patients with breast cancer, lymphoma, and lung cancer.

Inclusion Criteria:

• A diagnosis of dedifferentiated liposarcoma confirmed at MSKCC.
• Metastatic and/or locally advanced or locally recurrent disease that is not surgically resectable.
• All patients must have measurable disease as defined by RECIST 1.1. Patients must also have evidence of disease progression by RECIST 1.1 within 6 months of first dose of study drug.
• Any number of prior therapies (including none) is permitted. The last dose of systemic therapy (include targeted therapies) must have been given at least 2 4 weeks prior to initiation of therapy. Patients receiving BCNU or mitomycin C must have received their last dose of such therapy at least 6 weeks prior to initiation of therapy.
• Patients with brain metastasis that have been treated with definitive surgery or radiation and have been clinically stable for 3 months are eligible.
• Age ≥ 18 years.
• ECOG performance status 0 or 1
• Adequate organ and marrow function as defined below (ULN indicates institutional upper limit of normal):
• Absolute neutrophil count ≥ 1.510^9/L
• Hemoglobin ≥ 8.0 g/dL
• WBC ≥ 3.0 x 10^9/L
• Platelets ≥ 100 x 10^9/L
• Total bilirubin ≤ 1.5 x ULN except for patients with known Gilbert syndrome
• AST(SGOT)/ALT(SGPT) ≤ 3 x institutional ULN
• Creatinine ≤ 1.5 x ULN or Creatinine Clearance > 50 mL/min (calculated by Cockcroft-Gault method)
• Patients must not have current evidence of another malignancy that requires treatment.
• Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence). Women must not breast feed while on study.
• Ability to understand and the willingness to sign a written informed consent document.
• Ability to swallow capsules

Exclusion Criteria:

• Patients who have not recovered from adverse events of prior therapy to ≤ NCI CTCAEv4.0 Grade 1.
• Patients receiving any other investigational agents.
• Patients who have received prior treatment with a selective CDK4 inhibitor
• Uncontrolled intercurrent illness including, but not limited to, known ongoing or active infection, including HIV, active hepatitis B or C, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia (specifically, atrial fibrillation or ventricular dysrhythmias except ventricular premature contractions), or psychiatric illness/social situations that would limit compliance with study requirements.
• Pregnant women and women who are breast-feeding.